
RH-\(\mathcal {Z}\) 1    \(\mathcal {Z}\)  is holomorphic in \(\left(\Gamma \cup \lbrace 0\rbrace  \right)\) .

RH-\(\mathcal {Z}\) 2    \(\mathcal {Z}_+(z;n)=\mathcal {Z}_-(z;n)J_{\mathcal {Z}}(z)\) ,  where
\(J_{\mathcal {Z}}(z)={\left\lbrace \begin{array}{ll}J_{\mathcal {X},(z), & z \in  \\J_{\mathcal {X},i}(z), & z\in \Gamma _{i}, \\J_{\mathcal {X},o}(z), & z\in \Gamma _{o}.}\end{array}\right.}\item \textbf {RH-\mathcal {Z}3} \quad As z \rightarrow \infty  we have\begin{gather*}\mathcal {Z}(z;n)=\big (\displaystyle I+\mathcal {O}\big (z^{-1}\big )\big ) \begin{pmatrix}z^n & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & z^{-n} & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!.\end{gather*}\item \textbf {RH-\mathcal {Z}4} \quad As z \rightarrow 0 we have\begin{gather*}\mathcal {Z}(z;n)=P(n)(I+\mathcal {O}(z)) \begin{pmatrix}1 & 0 & 0 & 0 \\0 & z^{-n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^n\end{pmatrix}\!.\end{gather*}\) 
Remark 2.5 The term “opening of the lenses” is usually used to describe situations where the jump matrix on the added contours is exponentially close to the identity matrix for large values of the parameter \(n\) . The passage \(\Gamma \) , corresponding to the RH transformation \(\mathcal {X}\mapsto \mathcal {Z}\) , is clearly not of this type. However, our secondary opening of the lenses (the passage \(\Gamma \mapsto \Gamma _S\)  which corresponds to the RH transformation \(T \mapsto S\) ) in Section REF  is an example of a usual opening of the lenses.

Remark 2.6 The primary opening of the lenses is essential for the progression of the RH analysis in the following sections. This is due to a technical reason that will be elaborated at the end of next section. Since the structure of jump matrices is different in Section , we do not have an analogous step when the Hankel symbol is supported on \([a,b]\) , \(0<a<b<1\).

Normalization of behaviours at \(\mathbf {0}\)  and \({\infty }\)Normalization of behaviours at 0 and infty
Following the natural steps of Riemann–Hilbert analysis, we will normalize the behavior of \(\mathcal {Z}\)  at 0 and \(\infty \) ; to this end let us define
\(T(z;n):=\mathcal {Z}(z;n){\left\lbrace \begin{array}{ll}\begin{pmatrix}z^{-n} & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & z^{n} & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & |z|>1,\vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 0 & 0 \\0 & z^{n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^{-n}\end{pmatrix}\!, & |z|<1.\end{array}\right.}\) 
It is very important to note that in order to have a suitable Riemann–Hilbert analysis, the normalization of behaviors at 0 and \(\infty \)  can only be carried out only after the undressing \(\mathcal {X} \mapsto \mathcal {Z}\) ; this is due to technical reasons that will be further commented about at the end of this section. We have the following RHP for \(T\) :

RH-\(T\) 1    \(T\)  is holomorphic in \(\left( \Gamma _{i} \cup \Gamma _{o} \right)\) .

RH-\(T\) 2    \(T_+(z;n)=T_-(z;n)J_T(z;n)\) , where
\(J_T(z;n)= {\left\lbrace \begin{array}{ll}\widehat{J}(z;n), & z \in  \\J_{\mathcal {X},i}(z), & z\in \Gamma _{i}, \\J_{\mathcal {X},o}(z), & z\in \Gamma _{o},\end{array}\right.} \quad \mbox{where} \quad \widehat{J}(z;n) = \begin{pmatrix}z^n & 0 & 0 & -\phi (z) \\0 & z^n & \tilde{\phi }(z) & 0 \\0 & 0 & z^{-n} & 0 \\0 & 0 & 0 & z^{-n}\end{pmatrix}\!,\!\!\!\) 
and the matrices \(J_{\mathcal {X},i}\)  and \(J_{\mathcal {X},o}\)  are defined by (REF ).

RH-\(T\) 3    As \(z \rightarrow \infty \) , we have \(T(z;n)=\big (\displaystyle I+\mathcal {O}\big (z^{-1}\big )\big )\) .

We observe that for \(z \in , \) JT\( can be factorized as follows\begin{gather}\widehat{J}(z;n)= \begin{pmatrix}I_2 & 0_2 \\z^{-n} \Phi ^{-1}(z) & I_2\end{pmatrix} \begin{pmatrix}0_2 & \Phi (z) \\- \Phi ^{-1}(z) & 0_2\end{pmatrix} \begin{pmatrix}I_2 & 0_2 \\z^{n} \Phi ^{-1}(z) & I_2\end{pmatrix} \nonumber \\\phantom{\widehat{J}(z;n)}\equiv J_{T,o}(z;n)\overset{\circ }{J}(z)J_{T,i}(z;n),\end{gather}where \) 02\( and \) I2\( are respectively \) 2 2\( zero and identity matrices and\begin{gather*}\Phi (z)= \begin{pmatrix}0 & -\phi (z) \\\tilde{\phi }(z) & 0\end{pmatrix}\!.\end{gather*}Note that \) JT,i\( is exponentially close to the identity matrix for \) z\( inside of the unit circle and \) JT,o\( is exponentially close to the identity matrix for \) z\( outside of the unit circle.\) Now we are in a position to address Remark REF  in the previous section. Indeed, if one normalizes the behaviors at 0 and \(\infty \)  without the undressing transformation \(\mathcal {X} \mapsto \mathcal {Z}\) ; i.e., by directly defining the function \(\mathcal {T}\)  as
\(\mathcal {T}(z;n):= \mathcal {X}(z;n) {\left\lbrace \begin{array}{ll}\begin{pmatrix}z^{-n} & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & z^{n} & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & |z|>1, \vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 0 & 0 \\0 & z^{n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^{-n}\end{pmatrix}\!, & |z|<1.\end{array}\right.}\) 
Then the jump matrix \(J_{\mathcal {T}}:= \mathcal {T}^{-1}_-\mathcal {T}_+\)  on the unit circle would be
\(J_{\mathcal {T}}(z;n) = \begin{pmatrix}z^n & 0 & z^{n}\tilde{w}(z) & -\phi (z) \\0 & z^n & \tilde{\phi }(z) & -z^{-n}w(z) \\0 & 0 & z^{-n} & 0 \\0 & 0 & 0 & z^{-n}\end{pmatrix}\!,\) 
for which finding a factorization like () remains a challenge, mainly due to presence of the large parameter \(n\)  in the 13 and 24 elements of \(J_{\mathcal {T}}\) . This fact justifies the necessity of the undressing step \(\mathcal {X} \mapsto \mathcal {Z}\) . Indeed, due to the specific matrix structure of the jump matrices \(J_{\mathcal {X},i}\)  and \(J_{\mathcal {X},o}\)  they do not change
under the transformation (REF ).

The secondary opening of the lenses
The next Riemann–Hilbert transformation \(T \mapsto S\) , provides us with a problem with jump conditions on five contours where three jump matrices do not depend on \(n\)  and the other two converge exponentially fast to the identity matrix as \(n \rightarrow \infty \) . Let us define the function \(S\) , suggested by (), as
\(S(z;n):=T(z;n) \times {\left\lbrace \begin{array}{ll}J^{-1}_{T,i}(z;n), & z \in \Omega ^{\prime }_1, \\J_{T,o}(z;n), & z \in \Omega ^{\prime }_2, \\I, & z \in \Omega ^{\prime \prime }_1 \cup \Omega ^{\prime \prime }_2 \cup \Omega _0 \cup \Omega _{\infty },\end{array}\right.}\) 
where the regions \(\Omega ^{\prime }_1\) , \(\Omega ^{\prime }_2\) , \(\Omega ^{\prime \prime }_1\)  and \(\Omega ^{\prime \prime }_2\)  are shown in Fig. REF . We have the following Riemann–Hilbert problem for \(S\)

RH-\(S\) 1    \(S\)  is holomorphic in \(\left( \Gamma _{i} \cup \Gamma _{o} \cup \Gamma ^{\prime }_{i} \cup \Gamma ^{\prime }_{o} \right)\) .

RH-\(S\) 2    \(S_+(z;n)=S_-(z;n)J_S(z;n)\) , where
\(J_S(z;n)={\left\lbrace \begin{array}{ll}\overset{\circ }{J}(z), & z \in  \\J_{T,i}(z;n), & z \in \Gamma ^{\prime }_{i}, \\J_{T,o}(z;n), & z \in \Gamma ^{\prime }_{o}, \\J_{\mathcal {X},i}(z), & z \in \Gamma _{i}, \\J_{\mathcal {X},o}(z), & z \in \Gamma _{o}.\end{array}\right.}\) 

RH-\(S\) 3    As \(z \rightarrow \infty \) , we have \(S(z;n)=\displaystyle I+\mathcal {O}\big (z^{-1}\big )\) .

<FIGURE>In the usual way, we will first try to solve this Riemann–Hilbert problem by disregarding the jump matrices which depend on \(n\) , this solution is denoted by \(\overset{\circ }{S}\)  and will be referred to as the global parametrix. Once we construct the global parametrix, we will consider the small-norm Riemann–Hilbert problem for the ratio \(R:=S(\overset{\circ }{S})^{-1}\)  and discuss its solvability in the forthcoming sections.

The global parametrix and the model Riemann–Hilbert problem  for the pair \({(\phi ,w)}\)The global parametrix and the model Riemann–Hilbert problem for the pair \((phi,w)\)
The \(S\) -RHP reduces to the following Riemann–Hilbert problem for the global parametrix \(\overset{\circ }{S}\) , when we ignore the jump matrices which are exponentially close to the identity matrix:

RH-\(\overset{\circ }{S}\) 1    \(\overset{\circ }{S}\)  is holomorphic in \((\Gamma _{i} \cup \Gamma _{o})\) .

RH-\(\overset{\circ }{S}\) 2    \(\overset{\circ }{S}_+(z)=\overset{\circ }{S}_-(z)J_{\overset{\circ }{S}}(z)\) , where
\(J_{\overset{\circ }{S}}(z)={\left\lbrace \begin{array}{ll}\overset{\circ }{J}(z), & z \in  \\J_{\mathcal {X},i}(z), & z \in \Gamma _{i}, \\J_{\mathcal {X},o}(z), & z \in \Gamma _{o}. \\\end{array}\right.}\) 

RH-\(\overset{\circ }{S}\) 3    As \(z\rightarrow \infty \) , we have \(\overset{\circ }{S}(z)=\displaystyle I+\mathcal {O}\big (z^{-1}\big )\) .

And we finally dress the \(\overset{\circ }{S}\) -RHP to obtain a model problem for the global parametrix having jumps only on the unit circle. We define the function \(\Lambda \)  as
\(\Lambda (z) := \overset{\circ }{S}(z) \times {\left\lbrace \begin{array}{ll}J_{\mathcal {X},i}(z), & z \in \Omega _1, \\J^{-1}_{\mathcal {X},o}(z), & z \in \Omega _2, \\I, & z \in \Omega _{0} \cup \Omega _{\infty }.\end{array}\right.}\) 
Now we arrive at the following Riemann–Hilbert problem for \(\Lambda \)  that from now on we will refer to as the model Riemann–Hilbert problem for the pair \((\phi ,w)\):

RH-\(\Lambda \) 1    \(\Lambda \)  is holomorphic in \(.\)

RH-\(\Lambda \) 2    \(\Lambda _+(z)=\Lambda _-(z)J_{\Lambda }(z)\) , for \(z \in , where\begin{gather*}J_{\Lambda }(z) = \begin{pmatrix}0 & 0 & 0 & -\phi (z) \\\displaystyle -\frac{w(z)}{\phi (z)} & 0 & \displaystyle \tilde{\phi }(z) - \frac{w(z)\tilde{w}(z)}{\phi (z)} & 0 \\0 & \displaystyle -\frac{1}{\tilde{\phi }(z)} & 0 & 0 \\\displaystyle \frac{1}{\phi (z)} & 0 & \displaystyle \frac{\tilde{w}(z)}{\phi (z)} & 0\end{pmatrix}\!.\end{gather*}\)

RH-\(\Lambda \) 3    As \(z \rightarrow \infty \) , we have \(\Lambda (z)=\displaystyle I+\mathcal {O}\big (z^{-1}\big )\) .

The conditions on \(w\)  and \(\phi \)  which ensure the solvability of this model problem are not completely known and categorized at this point. We also want to stress that the appearance of the \(4\times 4\) 
model \(\Lambda \) -problem in the asymptotic analysis of the original \(\mathcal {X}\) -RHP is the crucial
difference of the Toeplitz+Hankel case we consider in this work comparing to the pure Toeplitz or
pure Hankel or Toeplitz+Hankel with the same symbols cases. Indeed, even if the pair \((\phi , w)\)  is such that
the \(\Lambda \) -RHP is solvable it does not mean that it is explicitly solvable. Hence, one should not
expect the closed form of the asymptotic answer in the case of the generic pair
\((\phi , w)\) .In this respect
the Toeplitz+Hankel determinants with generic symbols are similar to the block Toeplitz determinants, where
the explicit answers can be obtained only in two cases: (a) the Fourier expansion of the
corresponding matrix symbol is one side truncated or (b) one can produce an explicit Wiener–Hopf factorization of the
symbol. However, in Section  we will present a detailed analysis of this model problem for a specific family of pairs \((\phi , w)\)  within the broader class of Toeplitz and Hankel weights considered by E. Basor and T. Ehrhardt in [1]} for which the model \(\Lambda \) -problem is explicitly solvable.
Remarkably, we arrive at the same model \(\Lambda \) -Riemann–Hilbert problem, if we start with a Hankel weight supported on the interval \([a,b]\) , \(0<a<b<1\) . This will be shown in the next section.

Toeplitz+Hankel determinants: Hankel weight supported on the interval \({[a,b]}\) , \({0<a<b<1}\)Toeplitz+Hankel determinants: Hankel weight supported on the interval \([a,b{]}\) , \(0<a<b<1\)
In this section we consider the determinant (REF ) where \(w_k\)  exist and are given by (), with \(I=[a,b]\) , \(0<a<b<1\) . We further assume that \(w\)  does not have Fisher–Hartwig singularities (see [2]}, [3]}, [4]}, or [5]} for instances of Fisher–Hartwig singularities on the real line). Let us again assume that the symbol \(\phi \)  is of Szegő-type. The Riemann–Hilbert approach outlined in this section can be naturally extended to the three other cases: i) \(-1<a<b<0\) , ii) \(-\infty <a<b<-1\) , and iii) \(1<a<b<\infty \) .
We consider the system of orthogonal polynomials \(\lbrace P_n(z)\rbrace \) , \(\deg P_n(z)=n\) , satisfying the following orthogonality conditions
\(\int ^{b}_{a} P_n(x)x^{k+s} w(x)\, {\rm d}x + \int _{ P_n(z)z^{-k-r}\phi (z) \frac{{\rm d}z}{2\pi {\rm i} z} = h_n \delta _{n,k}, \qquad k=0,1,\dots , n.}One can write a determinantal formula for P_n like in (\ref {T+H OP Det rep}), which yields\begin{gather*}h_n=\frac{D_{n+1}(\phi ,w;r,s)}{D_{n}(\phi ,w;r,s)}.\end{gather*}By similar considerations as those mentioned in Section~\ref {Section Hankel on T}, the orthogonal polynomials P_n exist and are unique if D_n \ne 0. Now, assuming that D_n, D_{n-1}\ne 0, we consider the function Y defined as\begin{gather}Y(z;n)\!=\!\begin{pmatrix}P_n(z) & \displaystyle \int ^{b}_{a} \frac{P_n(x)x^sw(x)}{x-z}{\rm d}x + \int _{ \frac{ \tilde{\phi }(\xi )\xi ^{r}\tilde{P}_n(\xi )}{\xi -z} \frac{{\rm d}\xi }{2\pi {\rm i} \xi }\\[2ex]-\displaystyle \frac{P_{n-1}(z)}{h_{n-1}} & -\displaystyle \frac{1}{h_{n-1}} \Bigg \lbrace \int ^{b}_{a} \frac{P_{n-1}(x)x^sw(x)}{x-z}{\rm d}x+ \!\! \int _{ \!\frac{ \tilde{\phi }(\xi )\xi ^r\tilde{P}_{n-1}(\xi )}{\xi -z} \frac{{\rm d}\xi }{2\pi {\rm i} \xi } \Bigg \rbrace }\!,}built from the orthogonal polynomials P_n satisfying (\ref {T+H2 Orthogonality}). Consider the following Riemann--Hilbert problem for finding the 2 \times 2 matrix Y satisfying\begin{itemize}\item \textbf {RH-Y1} \quad Y is holomorphic in ([a,b]).\item \textbf {RH-Y2} \quad For z \in  we have\begin{gather*}Y_+^{(1)}(z;n)=Y^{(1)}_-(z;n),\end{gather*}and\begin{gather*}Y_+^{(2)}(z;n)=Y^{(2)}_-(z;n) + z^{r-1}\tilde{\phi }(z)Y^{(1)}_-\big (z^{-1};n\big ).\end{gather*}\end{itemize}\item \textbf {RH-Y3} \quad For x \in (a,b) we have\begin{gather*}Y_+^{(1)}(x;n)=Y^{(1)}_-(x;n),\end{gather*}and\begin{gather*}Y_+^{(2)}(x;n)=Y^{(2)}_-(x;n) + 2\pi {\rm i} x^s w(x)Y^{(1)}_-(x;n).\end{gather*}\end{pmatrix}\item \textbf {RH-Y4} \quad As z \rightarrow \infty \begin{gather*}Y(z;n)=\bigg ( I + O\bigg (\frac{1}{z}\bigg ) \bigg ) z^{n \sigma _3} = \begin{pmatrix}z^n+\mathcal {O}\big (z^{n-1}\big ) & \mathcal {O}\big (z^{-n-1}\big ) \\\mathcal {O}\big (z^{n-1}\big ) & z^{-n} + \mathcal {O}\big (z^{-n-1}\big )\end{pmatrix}\!,\end{gather*}\end{gather}where Y^{(1)} and Y^{(2)} are the first and second columns of Y, respectively. We have the analogue of Theorem \ref {THM RHP formulation circle} here as well.\) 

Theorem 3.1 
The following statements are true.

Suppose that \(D_n,D_{n-1} \ne 0\) . Then, the Riemann–Hilbert problem RH-\(Y\) 1 through RH-\(Y\) 4 is uniquely solvable and its solution \(\mathcal {Y}\)  is defined by (). Moreover,
\(h_{n-1} = - \lim _{z \rightarrow \infty } z^{n-1}/Y_{21}(z; n).\) 


Suppose that the Riemann–Hilbert problem RH-\(Y\) 1 through RH-\(Y\) 4
has a unique solution. Then \(D_n \ne 0 \) , \(\operatorname{rank} (T_{n-1}[\phi ;r] + H_{n-1}[w;s]) \ge n-2\) , and \(P_n(z)= Y_{11}(z;n)\) .

Suppose that the Riemann–Hilbert problem RH-\(Y\) 1 through RH-\(Y\) 4
has a unique solution. Suppose also that
\(\lim _{z \rightarrow \infty }Y_{21}(z;n)z^{-n+1} \ne 0.\) 
Then, as before, \(D_n \ne 0\) , \(P_n(z)= Y_{11}(z;n)\) , and, in addition,
\(D_{n-1} \ne 0, \qquad h_{n-1} = -\lim _{z\rightarrow \infty } \mathcal {Y}^{-1}_{21}(z;n) z^{n-1},\qquad P_{n-1}(z) = -h_{n-1}Y_{21}(z;n).\) 



We omit the proof here as it is similar to the proof of Theorem .
Corollary 3.2  Suppose that the \(Y\) -RH problem has a unique solution for \(n\)  and \(n-1\) .
Then
\(D_n \ne 0, \qquad D_{n-1} \ne 0, \qquad \mbox{and} \qquad h_{n-1} \ne 0.\) 
Moreover,
\(h_{n-1} = - \lim _{z \rightarrow \infty } z^{n-1}/Y_{21}(z;n).\) 


The associated \({2\times 4}\)  and \({4 \times 4}\)  Riemann–Hilbert problemsThe associated 2 x 4 and 4 x 4 Riemann–Hilbert problems
The formulation of the \(2\times 4\)  and \(4\times 4\)  Riemann–Hilbert problems are very similar to those of Section , however there are minor differences that convinces us to practice clarity in our exposition. Let us consider the following \(2\times 4\)  matrix function, constructed from the columns of \(Y\)  given by ():
<FIGURE>\(\overset{\circ }{X}(z;n) :=\big (Y^{(1)}(z;n), \widetilde{Y}^{(1)}(z;n), Y^{(2)}(z;n), \widetilde{Y}^{(2)}(z;n)\big ).\) 
Let us define \(\Sigma := [a,b] \cup \big [b^{-1},a^{-1}\big ]\) , and \(\Sigma ^{\prime }:=\Sigma \setminus \big \lbrace a,b,b^{-1},a^{-1}\big \rbrace \) . \(\overset{\circ }{X}(z;n)\)  satisfies the following Riemann–Hilbert problem

RH-\(\overset{\circ }{X}\) 1       \(\overset{\circ }{X}\)  is analytic in
\(\left( \Sigma \cup \lbrace 0\rbrace  \right)\) ,

RH-\(\overset{\circ }{X}\) 2       For \(z\in \Sigma ^{\prime } \) , we have \(\overset{\circ }{X}_+(z;n)=\overset{\circ }{X}_-(z;n)J_{\overset{\circ }{X}}(z)\) , where
\(J_{\overset{\circ }{X}}(z) = {\left\lbrace \begin{array}{ll}\begin{pmatrix}1 & 0 & 0 & -z^{-r+1}\phi (z) \\0 & 1 & z^{r-1}\tilde{\phi }(z) & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & z \in \vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 2\pi {\rm i}x^sw(x) & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & z\equiv x \in (a,b),\vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & -2 \pi {\rm i} x^{-s} \tilde{w}(x) \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & z\equiv x \in \big (b^{-1},a^{-1}\big ).\end{array}\right.}\) 


RH-\(\overset{\circ }{X}\) 3       As \(z\rightarrow \infty \)
\(\overset{\circ }{X}(z;n) =\begin{pmatrix}1 + O\big (z^{-1}\big ) & E_1(n)+O\big (z^{-1}\big ) & O\big (z^{-1}\big ) & E_3(n) + O\big (z^{-1}\big ) \vspace{2.84526pt}\\O\big (z^{-1}\big ) & E_2(n)+O\big (z^{-1}\big ) & 1 + O\big (z^{-1}\big ) & E_4(n) + O\big (z^{-1}\big )\end{pmatrix}\\\phantom{\overset{\circ }{X}(z;n) =}\times \begin{pmatrix}z^n & 0 & 0 & 0\\0 & 1 & 0 & 0 \\0 & 0 & z^{-n} & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!.\) 


RH-\(\overset{\circ }{X}\) 4       As \(z\rightarrow 0\)
\(\overset{\circ }{X}(z;n) \!=\! \begin{pmatrix}E_1(n) + O(z) & 1 + O(z) & E_3(n)+O(z) & O(z) \\E_2(n) + O(z) & O(z) & E_4(n)+O(z) & 1 + O(z)\end{pmatrix}\begin{pmatrix}1 & 0 & 0 & 0\\0 & z^{-n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^n\end{pmatrix}\!,\) 


where 3
 E1(n)= Y11(0;n),       
E3(n)= Y12(0;n),
 E2(n)= Y21(0;n),       
E4(n)= Y22(0;n). 

It is straightforward to check that \(\overset{\circ }{X}\)  given by (REF ) and () satisfies the Riemann–Hilbert problem RH-\(\overset{\circ }{X}\) 1 through RH-\(\overset{\circ }{X}\) 4.
One of the differences between the case when the Hankel symbol is supported on the unit circle versus the case when it is supported on the interval \([a,b]\) , is discussed in the following remark about the values of offsets that can be handled without much difficulty in each case.
Remark 3.3 
Let \(u\)  be defined by
\(u(z):= z\int ^b_{a} \frac{t^{s-1}w(t)}{t-z}{\rm d}t.\) 
When the Hankel symbol is supported on the interval, the natural progression of the Riemann–Hilbert analysis with general offset values \(r, s \in {\mathbb {Z}}\) , finally requires us to construct Szegő functions for the functions \(f_1(z)=z^{r-1}\phi (z)\)  and \(f_2(z)=\tilde{u}(z)\)  (Compare with Remark ). Note that the function \(u\)  given by (REF ) (and \(\tilde{u}\) ) has no winding number for all \(s \in {\mathbb {Z}}\) . Therefore, in this work it seems natural for us to focus on the determinants of the type \(D_n(\phi ,w;1,s)\) , in view of the points mentioned above and in Remark .

Similar to our approach in Section , we introduce the following Riemann–Hilbert problem of finding the \(4 \times 4\)  matrix function \(X\)  satisfying:

RH-X1       \(X\)  is analytic in
\((\Sigma \cup \lbrace 0\rbrace )\) .

RH-X2       For \(z\in \Sigma ^{\prime }\) , we have \(X_+(z;n)=X_-(z;n)J_{X}(z)\) , where
\(J_{X}(z) = {\left\lbrace \begin{array}{ll}\begin{pmatrix}1 & 0 & 0 & -\phi (z) \\0 & 1 & \tilde{\phi }(z) & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & z \in \vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 2\pi {\rm i}x^sw(x) & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & z\equiv x \in (a,b), \vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & -2 \pi {\rm i} x^{-s} \tilde{w}(x) \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & z\equiv x \in \big (b^{-1},a^{-1}\big ),\end{array}\right.}\) 
that is, \(J_X\)  is exactly equal to \(J_{\overset{\circ }{X}}\)  when \(r=1\) .

RH-X3       As \(z\rightarrow \infty \)
\(X(z;n) = \big (I+{\cal O}{\big (z^{-1}\big )}\big )\begin{pmatrix}z^n & 0 & 0 & 0\\0 & 1 & 0 & 0 \\0 & 0 & z^{-n} & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!.\) 

RH-X4       As \(z\rightarrow 0\)
\(X(z;n) =Q(n) (I+{\cal O}{(z)})\begin{pmatrix}1 & 0 & 0 & 0\\0 & z^{-n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^n\end{pmatrix}\!,\) 


where we emphasize that the matrix factor \(Q(n)\)  in (REF ) is not a priori prescribed. Now we are going to briefly mention some facts about this Riemann–Hilbert problem which are similar to those of the \(\mathcal {X}\) -RHP. Using the usual
Liouville theorem-based arguments one can easily show that the solution of \(X\) -RHP is unique, if it exists. Also, without much difficulty one can show that the function \(WP^{-1}(n)X\big (z^{-1};n\big )W\)  is also a solution of the \(X\) -RHP, and thus due to the uniqueness of the solution, we get the symmetry relation
\(WP^{-1}(n)\mathcal {X}\big (z^{-1};n\big )W = \mathcal {X}(z;n),\) 
where \(W\)  is given by (). Equation (REF ) yields the following symmetry equation for \(Q(n)\) ,
\(Q(n) = WQ^{-1}(n)W,\) 
or,
\((WQ(n))^2=(Q(n)W)^2 = I_4.\) 
Exact similar argument used in Remark  proves that the matrix \(Q(n)W-I_4\)  has rank 2. Here we also have the relationship between \(\overset{\circ }{X}\)  and \(X\)  given by
\(\overset{\circ }{X}(z;n)=\begin{pmatrix}1 & E_1(n) & 0 & E_3(n) \\0 & E_2(n) & 1 & E_4(n)\end{pmatrix}X(z;n),\) 
and moreover,
\(\begin{pmatrix}1 & E_1(n) & 0 & E_3(n) \\0 & E_2(n) & 1 & E_4(n)\end{pmatrix} = \begin{pmatrix}E_1(n) & 1 & E_3(n) & 0 \\E_2(n) & 0 & E_4(n) & 1\end{pmatrix} Q^{-1}(n).\) 
Since this is exactly the system (REF ), where \(C_j(n)\)  and \(P(n)\)  are respectively replaced by \(E_j(n)\)  and \(Q(n)\) , \(1\le j \le 4\) , and because \(P(n)\)  and \(Q(n)\)  enjoy the same symmetry and rank properties, we readily have the following statements, whose counterparts are already proven in Section .
Lemma 3.4 
Let \(Q_{jk}(n)\) , \(1 \le j,k \le 4\) , be the entries of the matrix \(Q(n)\) . Assume that at least one of the following six inequalities is true,
\(Q_{22}(n)Q_{44}(n) - Q_{42}(n)Q_{24}(n) \ne 0, \\(1-Q_{21}(n))Q_{42}(n) + Q_{22}(n)Q_{41}(n) \ne 0, \\(1-Q_{43}(n))Q_{22}(n) + Q_{23}(n)Q_{42}(n) \ne 0,\\(1-Q_{21}(n))Q_{44}(n) +Q_{41}(n)Q_{24}(n) \ne 0,\\(1-Q_{21}(n))(Q_{43}(n) - 1) +Q_{41}(n)Q_{23}(n) \ne 0,\\(1-Q_{43}(n))Q_{24}(n) + Q_{23}(n)Q_{44}(n) \ne 0.\) 
Then, the system (REF ) is a well-defined linear system on \(E_j(n)\)  which is uniquely solvable.

Lemma 3.5 
If the system (REF ) has a solution, it has to be unique.

Lemma 3.6 
Suppose that the solution of the \(X\) -RHP exists. Then, if at least one of the conditions (REF ) through () holds, one can uniquely reconstruct the solution of the \(Y\) -RHP.

Corollary 3.7 Suppose that the solution of the \(X\) -RHP exists for \(n\)  and \(n-1\) , then if at least one of the conditions (REF ) through () also holds for \(n\)  and \(n-1\) , then we have
\(D_n \ne 0, \qquad D_{n-1}\ne 0, \qquad \mbox{and} \qquad h_{n-1}\ne 0,\) 
where \(h_{n-1}\)  can be reconstructed form the RHP data as
\(h_{n-1} = - \lim _{z \rightarrow \infty } z^{n-1}/Y_{21}(z;n).\) 



Normalization of behaviors at 0 and \({\infty }\)Normalization of behaviors at 0 and infty
Unlike the situation in Section  where we had to make the transformation \(\mathcal {X} \mapsto \mathcal {Z}\)  before normalization of behaviors at zero and infinity, when the Hankel symbol is supported on the interval \([a,b]\)  we can immediately normalize the asymptotic behaviors at 0 and infinity, due to the desired structure of jump matrices. Indeed, it is natural to define
\(T(z;n):=X(z;n){\left\lbrace \begin{array}{ll}\begin{pmatrix}z^{-n} & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & z^{n} & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & |z|>1, \vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 0 & 0 \\0 & z^{n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^{-n}\end{pmatrix}\!, & |z|<1.\end{array}\right.}\) 
The function \(T\)  satisfies the following Riemann–Hilbert problem:

RH-\(T\) 1    \(T\)  is holomorphic in \(\Sigma \) .

RH-\(T\) 2    For \(z\in \Sigma ^{\prime }\) , we have \(T_+(z;n)=T_-(z;n)J_T(z;n)\) , where
\(J_T(z;n)= {\left\lbrace \begin{array}{ll}\widehat{J}(z;n), & z \in  \\J_{X}(z), & z\in (a,b)\cup \big (b^{-1},a^{-1}\big ).\end{array}\right.}\) 
We recall that \(\widehat{J}\)  is given by (REF ) and the matrices \(J_{X}\)  for \(z\in (a,b)\)  and \(z \in \big (b^{-1},a^{-1}\big )\)  are given by (REF ).

RH-\(T\) 3    As \(z \rightarrow \infty \) , we have \(T(z;n)=\big (\displaystyle I+\mathcal {O}\big (z^{-1}\big )\big )\) .

We bring the reader's attention to the fact that the transformation (REF ) does not change the jump matrices \(J_{X}\) .

Opening of the lenses
Using (), we open the lenses off the unit circle as shown in the Fig. REF  and we define
\(S(z;n):=T(z;n) \times {\left\lbrace \begin{array}{ll}J^{-1}_{T,i}(z;n), & z \in \Omega _1, \\J_{T,o}(z;n), & z \in \Omega _2, \\I, & z \in \left( \overline{\Omega _1} \cup \overline{\Omega _2} \cup [a,b] \cup \big [b^{-1},a^{-1}\big ] \right),\end{array}\right.}\) 
where \(J_{T,i}\)  and \(J_{T,o}\)  are defined in (). Let \(\Sigma _S \equiv \Sigma \cup \Sigma _o \cup \Sigma _i\)  and \(\Sigma ^{\prime }_S \equiv \Sigma ^{\prime } \cup \Sigma _o \cup \Sigma _i\)  (see Fig. REF ). It is straightforward to check that \(S\)  satisfies the following Riemann–Hilbert problem
<FIGURE>
RH-\(S\) 1    \(S\)  is holomorphic in \(\Sigma _S\) ,

RH-\(S\) 2    For \(z \in \Sigma ^{\prime }_S\)  we have \(S_+(z;n)=S_-(z;n)J_S(z;n)\) , where
\(J_S(z;n)={\left\lbrace \begin{array}{ll}\overset{\circ }{J}(z), & z \in  \\J_{T,i}(z;n), & z \in \Sigma _{i}, \\J_{T,o}(z;n), & z \in \Sigma _{o}, \\J_{X}(z), & z \in (a,b) \cup \big (b^{-1},a^{-1}\big ), \\\end{array}\right.}\) 
where these matrices are defined in () and (REF ).

RH-\(S\) 3    As \(z \rightarrow \infty \) , we have \(S(z;n)=\displaystyle I+\mathcal {O}\big (z^{-1}\big )\) .


The global parametrix and a model Riemann–Hilbert problem
Let us consider the following Riemann–Hilbert for \(\overset{\circ }{S}\) , or the global parametrix, which is expected to be a good approximation to \(S\)  for large parameter \(n\) . This RHP is simply obtained from the \(S\) -RHP by ignoring the jumps on \(\Sigma _i\)  and \(\Sigma _o\) :

RH-\(\overset{\circ }{S}\)1    \(\overset{\circ }{S}\)  is holomorphic in \(\Sigma \)  (see Fig. REF ).

RH-\(\overset{\circ }{S}\) 2    For \(z\in \Sigma ^{\prime }\) , we have
\(\overset{\circ }{S}_+(z)=\overset{\circ }{S}_-(z)J_{\overset{\circ }{S}}(z)\) , where
\(J_{\overset{\circ }{S}}(z)={\left\lbrace \begin{array}{ll}\overset{\circ }{J}(z), & z \in  \\J_{X}(z), & z \in (a,b) \cup \big (b^{-1},a^{-1}\big ). \\\end{array}\right.}\) 

RH-\(\overset{\circ }{S}\) 3    As \(z \rightarrow \infty \) , we have \(\overset{\circ }{S}(z)=\displaystyle I+\mathcal {O}\big (z^{-1}\big )\) .

Let us recall the function \(u\)  defined in Remark REF :
\(u(z)= z\int ^b_{a} \frac{t^{s-1}w(t)}{t-z}\,{\rm d}t.\) 
The Plemelj–Sokhotskii formula implies that
\(u_+(x)-u_-(x)=2\pi {\rm i} x^sw(x), \qquad x \in (a,b),\\\tilde{u}_+(x)-\tilde{u}_-(x)= -2\pi {\rm i} x^{-s}\tilde{w}(x), \qquad x \in \big (b^{-1},a^{-1}\big ).\) 
Put
\(\Theta (z):= \overset{\circ }{S}(z) {\left\lbrace \begin{array}{ll}\begin{pmatrix}1 & 0 & -u(z) & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & |z|<1, \vspace{2.84526pt}\\\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & -\tilde{u}(z) \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix}\!, & |z|>1.\end{array}\right.}\) 
It can be checked that \(\Theta \)  does not have jumps on the intervals \((a,b)\)  and \(\big (b^{-1},a^{-1}\big )\) . We have arrived at the following model Riemann–Hilbert problem on the unit circle:

RH-\(\Theta \) 1    \(\Theta \)  is holomorphic in \(.\)

RH-\(\Theta \) 2    \(\Theta _+(z)=\Theta _-(z)J_{\Theta }(z)\) , for \(z \in , where\begin{gather*}J_{\Theta }(z) = \begin{pmatrix}0 & 0 & 0 & -\phi (z) \\\displaystyle \frac{\tilde{u}(z)}{\phi (z)} & 0 & \displaystyle \tilde{\phi }(z) - \frac{u(z)\tilde{u}(z)}{\phi (z)} & 0 \\0 & \displaystyle -\frac{1}{\tilde{\phi }(z)} & 0 & 0 \\\displaystyle \frac{1}{\phi (z)} & 0 & \displaystyle -\frac{u(z)}{\phi (z)} & 0\end{pmatrix}\!.\end{gather*}\)

RH-\(\Theta \) 3    As \(z \rightarrow \infty \) , we have \(\Theta (z)=\displaystyle I+\mathcal {O}\big (z^{-1}\big )\) ,

where, in RH-\(\Theta \) 3 we have used the fact that \(\tilde{u}(\infty )=u(0)=0\) .
Remark 3.8 
Recalling Section REF , we note that this is exactly the model Riemann–Hilbert problem for the pair \((\phi ,-\tilde{u})\) . Hence, it can be concluded that the study of the Toeplitz+Hankel determinants both when the Hankel symbol is supported on the unit circle and also when it is supported on the interval \([a,b]\) , reduces to the study of the model Riemann–Hilbert problem RH-\(\Lambda \) 1 through RH-\(\Lambda \) 3.

For a specific class of symbols \(\phi \)  and \(w\) , we will present the solution to the model Riemann–Hilbert problem RH-\(\Lambda \) 1 through RH-\(\Lambda \) 3 for the pair \((\phi ,w)\)  in the next section.

Analysis of the model problem and a solvable pair
As mentioned before, it is an ambitious task to classify all the pairs \((\phi ,w)\)  for which the \(\Lambda \) -model Riemann–Hilbert problem is solvable. However, it is reasonable to start our analysis with the class of symbols () considered in [1]}. Since in our work the symbols are not assumed to be of the Fisher–Hartwig type (which needs a more delicate treatment, see Section REF ), we should still expect that the model Riemann–Hilbert problem be solvable for the class of symbols () when there is no Fisher–Hartwig singularity(\(a_0(z)=b_0(z) \equiv 1\) ). Indeed this is the case as will be elaborated in this section. As commented in the beginning of Section , asymptotics of \(D_n(\phi ,d\phi ;r,s)\) , for general \(r\)  and \(s\)  requires a more delicate approach (see Section REF ) and we do not discuss the details here. So let us consider \(D_n(\phi ,d\phi ;1,1)\) , where \(d\)  is of Szegő-type and further satisfies the condition \(d(z)\tilde{d}(z)=1\)  on the unit circle. For instance, a class of functions satisfying these conditions is given by
\(d(z)= \prod ^{m}_{j=1} d_j(z), \qquad d_j(z)=\pm \bigg (\frac{z-b_j}{z-a_j}\bigg )^{\alpha _j}\bigg (\frac{a_{j}z-1}{b_{j}z-1}\bigg )^{\alpha _j},\) 
where \(\alpha _j\in , all factors are defined by their principal branch, and\begin{gather*}0<a_1<b_1<a_2<b_2< \cdots < a_m< b_m < 1.\end{gather*}Note that a similar construction can be found for \) -1< bm < am < < b1< a1 < 0\(, and thus a larger class of functions can be found from multiplying functions of the first class with those of the second class. Although we have a class of functions satisfying the required properties, a complete categorization of functions satisfying the four required properties for \) d\( is yet to be found. We emphasize that the conditions \) d(1)=1\( required in~\cite {BE} do not play a role in the Riemann--Hilbert analysis. Indeed, for \) d\( as defined in (\ref {T+H Def d}) one can check that \) d(1)=(-1)0\(,where \) 0\( is the number of the \) dj\(-factors in whose definition the sign ``\) -\(^{\prime \prime } is taken. So in this sense we are considering functions \) d\( which are slightly more general than those considered in~\cite {BE}. At~the same time, we have our technical assumption of analyticityof the symbols in a~neighborhood of the unit circle which is not needed in the analysis of \cite {BE}.\)
Note that the condition \(d\tilde{d}=1\)  on the unit circle renders the 23-element of the jump matrix \(J_{\Lambda }\)  zero; indeed
\(J_{\Lambda ,23}(z) = \displaystyle \tilde{\phi }(z) - \frac{w(z)\tilde{w}(z)}{\phi (z)} = \tilde{\phi }(z) \big ( 1 - d(z) \tilde{d}(z)\big )=0.\) 
Hence, for the particular choices made above, the jump matrix \(G_{\Lambda }\)  reduces to
\(J_{\Lambda }(z) = \begin{pmatrix}0 & 0 & 0 & -\phi (z) \\\displaystyle - d(z) & 0 & 0 & 0 \\0 & \displaystyle -\frac{1}{\tilde{\phi }(z)} & 0 & 0 \\\displaystyle \frac{1}{\phi (z)} & 0 & \displaystyle \frac{\tilde{w}(z)}{\phi (z)} & 0\end{pmatrix}\!. \) 
In order to factorize \(J_{\Lambda }\) , let us first consider the following Szegő functions
\(\alpha (z)=\exp \bigg [ \frac{1}{2 \pi {\rm i} } \int _{ \frac{\ln (\phi (\tau ))}{\tau -z}\,{\rm d}\tau \bigg ], \qquad \beta (z)=\exp \bigg [ \frac{1}{2 \pi {\rm i} } \int _{ \frac{\ln (d(\tau ))}{\tau -z}\,{\rm d}\tau \bigg ].}By Plemelj--Sokhotskii formula \alpha , \beta , \tilde{\alpha } and \tilde{\beta } satisfy the following jump conditions on the unit circle:\begin{gather}\begin{split}& \alpha _+(z)=\alpha _-(z) \phi (z), \qquad \beta _+(z)=\beta _-(z) d(z), \\ & \tilde{\alpha }_-(z)=\tilde{\alpha }_+(z) \tilde{\phi }(z), \qquad \tilde{\beta }_-(z)=\tilde{\beta }_+(z) \tilde{d}(z).\end{split}\end{gather}It turns out that knowing the value of \beta (0) is crucial for finding an asymptotic expression for~h_n (see Section~\ref {Section T+H Asymp h_n}) and the condition d\tilde{d}=1 on the unit circle allows us to evaluate \beta (0) easily. Indeed\begin{gather*}\int _{\ln (d(\tau )) \,\frac{{\rm d}\tau }{\tau } = \int _{\ln \big (\tilde{d}(\tau )\big ) \,\frac{{\rm d}\tau }{\tau }= \int _{\ln \big (d^{-1}(\tau )\big ) \,\frac{{\rm d}\tau }{\tau }= - \int _{\ln (d(\tau )) \,\frac{{\rm d}\tau }{\tau }.}Thus\begin{gather}\int _{\ln (d(\tau )) \frac{{\rm d}\tau }{\tau }=0, \qquad \mbox{and therefore,} \qquad \beta (0)=1.}Next, we show that \beta =\tilde{\beta }. Note that\begin{gather*}\begin{split}{\beta }(z) & = \exp \bigg [ \frac{1}{2 \pi {\rm i} } \int _{ \frac{\ln (d(\tau ))}{\tau -z^{-1}}\,{\rm d}\tau \bigg ] = \exp \bigg [ -\frac{z}{2 \pi {\rm i} } \int _{ \frac{\ln (d(\tau ))}{\tau ^{-1}-z} \,\frac{{\rm d}\tau }{\tau } \bigg ]\\& = \exp \bigg [ -\frac{z}{2 \pi {\rm i} } \int _{ \frac{\ln \big ({d}(\tau )\big )}{\tau -z} \frac{{\rm d}\tau }{\tau } \bigg ] = \exp \bigg [ \frac{z}{2 \pi {\rm i} } \int _{ \frac{\ln (d(\tau ))}{\tau -z} \,\frac{{\rm d}\tau }{\tau } \bigg ],}}where we have again used the fact that d\tilde{d}\equiv 1 on the unit circle. Using\begin{gather*}\frac{1}{(\tau -z)\tau } = \frac{z^{-1}}{\tau -z} - \frac{z^{-1}}{\tau },\end{gather*}we can write the last expression for \tilde{\beta } as\begin{gather*}{\beta }(z) = \exp \bigg [ \frac{1}{2 \pi {\rm i} } \int _{ \frac{\ln (d(\tau ))}{\tau -z}\,{\rm d}\tau - \frac{1}{2 \pi {\rm i} } \int _{ \frac{\ln (d(\tau ))}{\tau } \, {\rm d}\tau \bigg ] = \frac{\beta (z)}{\beta (0)} = \beta (z),}by (\ref {T+H be0}). To show that \beta =\tilde{\beta } one could also argue that they both solve \textit {the same} scalar RHP which has a unique solution. We also note that \alpha (z),\beta (z) = 1 + \mathcal {O}(z^{-1}), and \tilde{\alpha }(z)= \alpha (0)(1+\mathcal {O}(z^{-1})) as z \rightarrow \infty . Now we can write the solution of the \Lambda -RHP (in the case d\tilde{d}\equiv 1 on ) as\begin{gather}\Lambda (z)= \Lambda ^{-1}_{\infty }\!\begin{pmatrix}1 & 0 & 0 & 0 \\\mathcal {C}_{\rho }(z) & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{pmatrix} \displaystyle \!\times \!{\left\lbrace \begin{array}{ll}\begin{pmatrix}\displaystyle -\beta (z) & 0 & 0 & 0 \\0 & 0 & \displaystyle \frac{1}{\tilde{\alpha }(z)\beta (z) \alpha (z)} & 0 \\0 & \displaystyle -\tilde{\alpha }(z) & 0 & 0 \\0 & 0 & 0 & \displaystyle -\alpha (z)\end{pmatrix}\!, & |z|<1, \\\begin{pmatrix}0 & \beta (z) & 0 & 0 \\0 & 0 & 0 & \displaystyle \frac{1}{\beta (z)\tilde{\alpha }(z)\alpha (z)} \\0 & 0 & \tilde{\alpha }(z) & 0 \\\alpha (z) & 0 & 0 & 0\end{pmatrix}\!, & |z|>1,\end{array}\right.}\hspace{-28.45274pt}\end{gather}where \mathcal {C}_f(z) is the Cauchy-transform of f(z):\begin{gather*}\mathcal {C}_f(z) = \frac{1}{2\pi {\rm i} } \int _{ \frac{f(\tau )}{\tau -z}\, {\rm d}\tau ,}and\begin{gather}\Lambda ^{-1}_{\infty } = \begin{pmatrix}0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 0 & \displaystyle \frac{1}{\alpha (0)} & 0 \\0 & \alpha (0) & 0 & 0\end{pmatrix}\!, \qquad \rho (z) = \displaystyle -\frac{1}{\beta _-(z) \beta _+(z) \tilde{\alpha }_-(z) \alpha _+(z) }.\end{gather}Using (\ref {T+H jumps of Szego functions}), the Plemelj--Sokhotskii formula and general properties of the Cauchy integral, it~can be checked that \Lambda  given by (\ref {Lambda}) satisfies the \Lambda -RHP.\end{gather*}\subsection [The small-norm Riemann--Hilbert problem associated to D_n(phi,dphi,1,1)]{The small-norm Riemann--Hilbert problem associated to {D_n(\phi ,d\phi ,1,1)}}}Let us consider\begin{gather*}R(z;n):= S(z;n) \overset{\circ }{S}(z)^{-1}.\end{gather*}This function clearly has no jumps on \Gamma _i, \Gamma _o and , since S and \overset{\circ }{S} have the same jumps on these contours. Thus, R satisfies the following small-norm Riemann--Hilbert problem\end{gather*}\begin{itemize}\item \textbf {RH-R1} \quad R is holomorphic in \Gamma _R.\end{itemize}\item \textbf {RH-R2} \quad R_+(z;n)=R_-(z;n)J_{R}(z;n), for z \in \Gamma _R.\item \textbf {RH-R3} \quad As z \rightarrow \infty , R(z;n)= I + \mathcal {O}\big (z^{-1}\big ),}where \Gamma _R := \Gamma _i^{\prime } \cup \Gamma _o^{\prime }, and J_R is given by\begin{gather*}J_R(z;n) = \overset{\circ }{S}(z) J_S(z;n) \overset{\circ }{S}(z)^{-1} = {\left\lbrace \begin{array}{ll}\overset{\circ }{S}(z) J_{T,i}(z;n) \overset{\circ }{S}(z)^{-1}, & z \in \Gamma _i^{\prime }, \\\overset{\circ }{S}(z) J_{T,o}(z;n) \overset{\circ }{S}(z)^{-1}, & z \in \Gamma _o^{\prime }.\end{array}\right.}\end{gather*}Using (\ref {Lambda}), (\ref {Lambda infty}), (\ref {9}), (\ref {GTfactorization}) and (\ref {GXfactorization}) we find\begin{gather}J_R(z;n)-I = {\left\lbrace \begin{array}{ll}z^n \begin{pmatrix}0 & g_{12}(z) & 0 & g_{14}(z) \\0 & 0 & g_{23}(z) & 0 \\0 & 0 & 0 & 0 \\0 & 0 & g_{43}(z) & 0\end{pmatrix}\!, & z \in \Gamma _i^{\prime }, \\z^{-n} \begin{pmatrix}0 & 0 & 0 & 0 \\g_{21}(z) & 0 & 0 & 0 \\0 & g_{32}(z) & 0 & g_{34}(z) \\g_{41}(z) & 0 & 0 & 0\end{pmatrix}\!, & z \in \Gamma _o^{\prime },\end{array}\right.}\end{gather}where\begin{gather*}g_{12}(z) = - \frac{\alpha (z)}{\phi (z)\beta (z)} - \frac{\tilde{w}(z) C_{\rho }(z)}{\phi (z)\beta (z)\tilde{\alpha }(z)}, \qquad g_{14}(z) = \frac{\tilde{w}(z)}{\phi (z)\beta (z)\tilde{\alpha }(z) \alpha (0) },\\g_{23}(z) = - \frac{\alpha (0) \tilde{w}(z) \beta (z) }{\tilde{\phi }(z)\tilde{\alpha }(z)}, \qquad g_{43}(z) = - \alpha ^2(0) \bigg ( \frac{\alpha (z) \beta (z)}{\tilde{\phi }(z)} + \frac{\beta (z)\tilde{w}(z) C_{\rho }(z)}{\tilde{\alpha }(z)\tilde{\phi }(z)} \bigg ),\\g_{21}(z) = \frac{w(z) \beta (z) }{\phi (z)\alpha (z)}, \qquad g_{32}(z) = - \frac{1}{\alpha (0)\tilde{\phi }(z)} \bigg ( \frac{\tilde{\alpha }(z)}{\beta (z)} - w(z)\tilde{\alpha }^2(z)\beta (z)\alpha (z)C_{\rho }(z) \bigg ),\\g_{34}(z) = \frac{w(z) \tilde{\alpha }^2(z)\beta (z)\alpha (z) }{\tilde{\phi }(z)\alpha ^2(0)}, \qquad g_{41}(z) = -\frac{\alpha (0)}{\phi (z)} \bigg ( \frac{1}{\tilde{\alpha }(z)\beta (z)\alpha ^2(z)} - \frac{w(z)\beta (z)C_{\rho }(z)}{\alpha (z)} \bigg ).\end{gather*}From (\ref {JR000}) it follows that the jump matrix J_R satisfies on \Gamma _Rthe small-norm estimate,\begin{gather}||J_R - I||_{L_{2} \cap L_{\infty }} \le C {\rm e}^{-cn},\end{gather}for some positive C and c = -\log r_1, where r_1 is any number satisfying the conditionr_0 < r_1 < 1 (see (\ref {annulus}) and (\ref {r0}) for the definition and meaning ofthe number r_0). Therefore, by standard theory of small-norm Riemann--Hilbert problems \cite {Deiftetal,Deiftetal2}, there exists n_* such that for all n > n_* the R-RH problem is solvable and\begin{gather*}R(z) = I + R_1(z) + R_2(z) + R_3(z) + \cdots , \qquad z \in \Gamma _R, \qquad n \ge n_*,\end{gather*}where each R_k is of order O\big ({\rm e}^{-kcn}\big ) and they can be found recursively from\begin{gather}R_k(z) = \frac{1}{2\pi {\rm i}}\int _{\Gamma _R} \frac{[ R_{k-1}(\mu )]_- (J_R(\mu )-I)}{\mu -z}{\rm d}\mu , \qquad z \in \Gamma _R, \qquad k\ge 1.\end{gather}Note that this recurrence also means that\begin{gather*}R_{k+1}(n) = o(R_k(n)),\qquad n \rightarrow \infty .\end{gather*}In particular,\begin{gather}R_1(z;n)=\frac{1}{2\pi {\rm i}}\int _{\Gamma _R} \frac{J_R(\mu ;n)-I}{\mu -z} \, {\rm d}\mu \nonumber \\\phantom{R_1(z;n)}= \begin{pmatrix}0 & R_{1,12}(z;n) & 0 & R_{1,14}(z;n) \\R_{1,21}(z;n) & 0 & R_{1,23}(z;n) & 0 \\0 & R_{1,32}(z;n) & 0 & R_{1,34}(z;n) \\R_{1,41}(z;n) & 0 & R_{1,43}(z;n) & 0\end{pmatrix}\!,\end{gather}where\begin{gather}\begin{split}& R_{1,jk}(z;n) = \frac{1}{2\pi {\rm i}}\int _{\Gamma ^{\prime }_i} \frac{\mu ^ng_{jk}(\mu )}{\mu -z}\,{\rm d}\mu , \qquad jk=12,14,23,43,\\& R_{1,jk}(z;n) = \frac{1}{2\pi {\rm i}}\int _{\Gamma ^{\prime }_o} \frac{\mu ^{-n}g_{jk}(\mu )}{\mu -z}\, {\rm d}\mu , \qquad jk=21,32,34,41.\end{split}\end{gather}Also from (\ref {Rk}) we can write an expression for R_2:\begin{gather*}R_2(z;n)= \frac{1}{2\pi {\rm i}}\int _{\Gamma _R} \frac{[ R_{1}(\mu ;n)]_- (J_R(\mu ;n)-I)}{\mu -z}\,{\rm d}\mu \\\phantom{R_2(z;n)}= \begin{pmatrix}R_{2,11}(z;n) & 0 & R_{2,13}(z;n) & 0 \\0 & R_{2,22}(z;n) & 0 & R_{2,24}(z;n) \\R_{2,31}(z;n) & 0 & R_{2,33}(z;n) & 0 \\0 & R_{2,42}(z;n) & 0 & R_{2,44}(z;n)\end{pmatrix}\!,\end{gather*}where\begin{gather*}R_{2,kj}(z;n) = {\left\lbrace \begin{array}{ll}\displaystyle \sum _{\ell \in \lbrace 2,4\rbrace } \frac{1}{2\pi {\rm i}} \int _{\Gamma ^{\prime }_o} \frac{\mu ^{-n} [R_{1,k\ell }(\mu ;n)]_-g_{\ell j}(\mu )}{\mu -z}\,{\rm d}\mu , \qquad j=1,\ k=1,3,\\\displaystyle \sum _{\ell \in \lbrace 2,4\rbrace } \frac{1}{2\pi {\rm i}} \int _{\Gamma ^{\prime }_i} \frac{\mu ^{n} [R_{1,k\ell }(\mu ;n)]_-g_{\ell j}(\mu )}{\mu -z}\,{\rm d}\mu , \qquad j=3,\ k=1,3,\\\displaystyle \frac{1}{2\pi {\rm i}} \int _{\Gamma ^{\prime }_i} \frac{\mu ^{n} [R_{1,k1}(\mu ;n)]_-g_{1j}(\mu )}{\mu -z}\,{\rm d}\mu \\ \phantom{\frac{1}{2\pi {\rm i}} \int _{\Gamma ^{\prime }_i}}\displaystyle +\frac{1}{2\pi {\rm i}} \int _{\Gamma ^{\prime }_o} \frac{\mu ^{-n} [R_{1,k3}(\mu ;n)]_-g_{3j}(\mu )}{\mu -z}\,{\rm d}\mu , \qquad k, j=2,4.\end{array}\right.}\end{gather*}Moreover, using (\ref {Rk}) and a straightforward calculation one can justify that the matrix structure (i.e., the location of zero and nonzero elements) of R_{2k+1} and R_{2k}, k\ge 1, are similar to that of~R_1 and R_2, respectively. It is also straightforward to show that\begin{gather*}R_{k,ij}(z;n) = \frac{{\cal O}{\big ({\rm e}^{-kcn}\big )}}{|z|+1}, \qquad n \rightarrow \infty , \qquad k\ge 1,\end{gather*}uniformly for z \in \Gamma _R, and the positive constant c is the same as in (\ref {smallnorn000}).}\subsection [Asymptotics of h{_}n]{Asymptotics of {h_n}}\end{split}The analysis of the previous section shows that the \mathcal {X}-RH problem has unique solution for alln > n_{*}. We now proceed to the reconstruction of the corresponding functions \overset{\circ }{\mathcal {X}}(z; n),\mathcal {Y}(z; n), and to the asymptotics of~h_n. To this end we need the asymptotic information aboutthe matrix~P(n) which is needed to determine constants C_j that participate in equation~(\ref {2 by 4 solution unique}).\end{gather*}Tracing back the Riemann--Hilbert transformations, we find that for z \in \Omega _0 we have\begin{equation}\mathcal {X}(z;n)=R(z;n)\Lambda (z)\begin{pmatrix}1 & 0 & 0 & 0\\0 & z^{-n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^{n}\end{pmatrix}\!.\end{equation}From (\ref {T+H P from X}) and (\ref {T+H Trace back X}) we conclude that \begin{equation}P(n) = R(0;n)\Lambda (0).\end{equation}From (\ref {R1000}) we arrive at the estimate\begin{gather}R(0;n) = I + R_1(0;n) +O\big ({\rm e}^{-2cn}\big ) \nonumber \\\phantom{R(0;n)}= \begin{pmatrix}1 & R_{1,12}(0;n) & 0 & R_{1,14}(0;n) \\R_{1,21}(0;n) & 1 & R_{1,23}(0;n) & 0 \\0 & R_{1,32}(0;n) & 1 & R_{1,34}(0;n) \\R_{1,41}(0;n) & 0 & R_{1,43}(0;n) & 1\end{pmatrix} + O\big ({\rm e}^{-2cn}\big ).\end{gather}Simultaneously, from (\ref {Lambda}) and (\ref {Lambda infty}) we have\begin{gather}\Lambda (0)= \begin{pmatrix}0 & 0 & 0 & -\alpha (0) \\-1 & 0 & 0 & 0 \\0 & -\displaystyle \frac{1}{\alpha (0)} & 0 & 0 \\-C_{\rho }(0)\alpha (0) & 0 & 1 & 0\end{pmatrix}\!.\end{gather}Equations (\ref {T+H P(n)}), (\ref {R0n000}) and (\ref {T+H LA(0)}) yield the followingasymptotic formula for P(n),\begin{gather}P(n)\!=\!\!\begin{pmatrix}-C_{\rho }(0)\alpha (0)R_{1,14}(0;\!n)\!-\!R_{1,12}(0;\!n) & 0 & R_{1,14}(0;\!n) & -\alpha (0)\\[1ex]-1 & -\displaystyle \frac{R_{1,23}(0;\!n)}{\alpha (0)} & 0 & -\alpha (0)R_{1,21}(0;\!n)\\[2ex]-C_{\rho }(0)\alpha (0)R_{1,34}(0;\!n)\!-\!R_{1,32}(0;\!n) & -\displaystyle \frac{1}{\alpha (0)} & R_{1,34}(0;\!n) & 0\\[2ex]-C_{\rho }(0)\alpha (0) & -\displaystyle \frac{R_{1,43}(0;\!n)}{\alpha (0)} & 1 & -\alpha (0)R_{1,41}(0;\!n)\nonumber \end{pmatrix}\\ \phantom{P(n)}+ {\cal O}{\big ({\rm e}^{-2cn}\big )},\end{gather}as n \rightarrow \infty .\end{gather}It is time now for the conditions of Lemma~\ref {Lemma Unique Solvability of C^{\prime }s}.We are not going to study each and every condition of this lemma, rather as a case studywe consider in particular the condition~(\ref {gencond}):\begin{gather}(1-P_{21}(n))P_{42}(n) + P_{22}(n)P_{41}(n) \ne 0.\end{gather}From (\ref {T+H P(n) Expansion}) we have that\begin{gather}(1-P_{21}(n))P_{42}(n) + P_{22}(n)P_{41}(n) = -\mathcal {E}(n) + O\big ({\rm e}^{-2cn}\big ), \qquad n \rightarrow \infty ,\end{gather}where (cf.~(\ref {En0001}))\begin{gather}\mathcal {E}(n):= \frac{2}{\alpha (0)}R_{1,43}(0;n)-C_{\rho }(0)R_{1,23}(0;n).\end{gather}This is when we arrive at condition (\ref {Enneq0}) of Theorem \ref {T+H main thm}. Indeed, let us suppose that there exist such C>0 and n_0 \ge n_{*} that\begin{gather*}|\mathcal {E}(n)| \ge Cr^{n}, \qquad \mbox{for some}\quad r\colon r_{0}\le r < 1, \qquad \mbox{and}\qquad n> n_0,\end{gather*}and choose r_1 in the definition of the constant c so that  r< r_1 <1 and r^{2}_1 < r. Then, estimate (\ref {condition22220}) can be rewritten as\begin{gather}(1-P_{21}(n))P_{42}(n) + P_{22}(n)P_{41}(n) = -\mathcal {E}(n)\big ( 1 + O\big ({\rm e}^{-c_1n} \big )\big ), \qquad n \rightarrow \infty ,\end{gather}where c_1 = -\log \big (\frac{r^2_1}{r}\big ) >0. The last estimate in turn means that there exists such \mbox{$n_1 \ge n_0$} that for all n > n_1 +1we should have that condition (\ref {condition 2222}) holds for n and n-1 andhence by~Lemma~\ref {Lemma Unique reconstruction of Y from X} and Corollary (\ref {corollary2.8.1}), we can uniquely reconstruct the solution of the \mathcal {Y}-RHP, having already the unique solution of the \mathcal {X}-RHP, and, moreover,we could use equation (cf.~(\ref {T+H h_n000})),\begin{gather}-\frac{1}{h_{n-1}} = \lim _{z \rightarrow 0} z^{n-1} \mathcal {Y}_{21}\big (z^{-1};n\big ).\end{gather}for evaluation of the large n behavior of h_n.Let us denote \begin{gather}\mathcal {A}(z;n) := P^{-1}(n)\mathcal {X}(z;n)\begin{pmatrix}1 & 0 & 0 & 0\\0 & z^{n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^{-n}\end{pmatrix}\!,\end{gather}and also let us define the matrix \mathcal {B}(n) in the following expansion for \mathcal {A}(z;n), which is equivalent to \textbf {RH-\mathcal {X}4}:\begin{gather}\mathcal {A}(z;n) = I + \mathcal {B}(n)z + {\cal O}{\big (z^2\big )}, \qquad z \rightarrow 0.\end{gather}Therefore by (\ref {24-to-44}), (\ref {mathfrakR}), (\ref {C^{\prime }s-to-P}) and (\ref {T+H A}) we can write\begin{gather}\overset{\circ }{\mathcal {X}}(z,n) = \begin{pmatrix}C_1(n) & 1 & C_3(n) & 0 \\C_2(n) & 0 & C_4(n) & 1\end{pmatrix} \mathcal {A}(z;n) \begin{pmatrix}1 & 0 & 0 & 0\\0 & z^{-n} & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & z^{n}\end{pmatrix}\!.\end{gather}Using (\ref {X naught to Y}) and (\ref {X-naught-good-equation}) we can write\begin{gather}\mathcal {Y}_{21}\big (z^{-1};n\big ) = \overset{\circ }{\mathcal {X}}_{22}(z;n) = C_2(n) \mathcal {A}_{12}(z;n)z^{-n} \!+ C_4(n)\mathcal {A}_{32}(z;n)z^{-n} \!+ \mathcal {A}_{42}(z;n)z^{-n}.\!\end{gather}From (\ref {T+H A exp zero}) we have\begin{gather*}z^{-n}\mathcal {A}(z;n) = z^{-n} I + z^{-n+1}\mathcal {B}(n) + \mathcal {O}\big (z^{-n+2}\big ), \qquad z \rightarrow 0.\end{gather*}Therefore, as z \rightarrow 0\begin{gather}z^{-n}\mathcal {A}_{ij}(z;n) ={\left\lbrace \begin{array}{ll}z^{-n+1}\mathcal {B}_{ij}(n) + \mathcal {O}\big (z^{-n+2}\big ), & i \ne j,\\z^{-n} + z^{-n+1}\mathcal {B}_{ii}(n) + \mathcal {O}\big (z^{-n+2}\big ), & i=j.\end{array}\right.}\end{gather}Therefore by (\ref {T+H h_n good}), (\ref {Y_21(z^-1) to C^{\prime }s}) and (\ref {T+H Asym A vs. B}) we have\begin{gather}-\frac{1}{h_{n-1}} = C_2(n) \mathcal {B}_{12}(n) + C_4(n)\mathcal {B}_{32}(n) + \mathcal {B}_{42}(n).\end{gather}Let us denote the coefficients in the expansions of R(z;n) and \Lambda (z), as z \rightarrow 0, by\begin{gather*}R(z;n) = R(0;n) + R^{(1)}(n) z + R^{(2)}(n) z^2 + {\cal O}{\big (z^3\big )},\\\Lambda (z) = \Lambda (0) + \Lambda ^{(1)} z + \Lambda ^{(2)} z^2 + {\cal O}{\big (z^3\big )}.\end{gather*}From (\ref {T+H A exp zero}), (\ref {T+H Trace back X}), and (\ref {T+H P(n)}) we have\begin{gather}\mathcal {B}(n) = \Lambda ^{-1}(0)R^{-1}(0;n)R^{(1)}(n)\Lambda (0) + \Lambda ^{-1}(0)\Lambda ^{(1)}.\end{gather}Note that\begin{gather*}R^{(1)}(n) = \frac{1}{2\pi {\rm i}} \int _{\Gamma _R}\! (J_R(\mu ;n)-I) \frac{{\rm d}\mu }{\mu ^2} + \mathcal {O}\big ({\rm e}^{-2cn}\big ), \qquad \!\!R^{-1}(0;n) = I - R_1(0;n) + {\cal O}{\big ({\rm e}^{-2cn}\big )},\end{gather*}as n \rightarrow \infty . More precisely, we have\begin{gather}R^{(1)}(n)= \begin{pmatrix}0 & R^{(1)}_{12}(n) & 0 & R^{(1)}_{14}(n) \\R^{(1)}_{21}(n) & 0 & R^{(1)}_{23}(n) & 0 \\0 & R^{(1)}_{32}(n) & 0 & R^{(1)}_{34}(n) \\R^{(1)}_{41}(n) & 0 & R^{(1)}_{43}(n) & 0\end{pmatrix} + \mathcal {O}\big ({\rm e}^{-2cn}\big ), \end{gather}as n \rightarrow \infty , where\begin{gather}\begin{split}& R^{(1)}_{jk}(n) = \frac{1}{2\pi {\rm i}}\int _{\Gamma ^{\prime }_i} \mu ^{n-2}g_{jk}(\mu )\,{\rm d}\mu , \qquad \ \ jk=12,14,23,43, \\& R^{(1)}_{jk}(n) = \frac{1}{2\pi {\rm i}}\int _{\Gamma ^{\prime }_o} \mu ^{-n-2}g_{jk}(\mu )\,{\rm d}\mu , \qquad jk=21,32,34,41,\end{split}\end{gather}and\begin{gather}R^{-1}(0;n)\!=\!\begin{pmatrix}1 & -R_{1,12}(0;n) & 0 & -R_{1,14}(0;n) \\-R_{1,21}(0;n) & 1 & -R_{1,23}(0;n) & 0 \\0 & -R_{1,32}(0;n) & 1 & -R_{1,34}(0;n) \\-R_{1,41}(0;n) & 0 & -R_{1,43}(0;n) & 1\end{pmatrix}\! + {\cal O}{\big ({\rm e}^{-2cn}\big )},\\ \qquad n \rightarrow \infty .\nonumber \end{gather}Regarding the coefficients of \Lambda (z), we will actually only needthe zero-term, \Lambda (0), which has already been presented in (\ref {T+H LA(0)}).}}From (\ref {T+H B}), (\ref {T+H R^(1)}), (\ref {T+H R-inverse at 0}) and (\ref {T+H LA(0)}) we find that\begin{gather}\mathcal {B}_{12}(n) = \frac{R^{(1)}_{23}(n)}{\alpha (0)} + \mathcal {O}\big ({\rm e}^{-2cn}\big ), \qquad \mathcal {B}_{32}(n) = C_{\rho }(0)R^{(1)}_{23}(n)-\frac{R^{(1)}_{43}(n)}{\alpha (0)} + \mathcal {O}\big ({\rm e}^{-2cn}\big ), \nonumber \\ \mathcal {B}_{42}(n) = - \frac{1}{\alpha ^2(0)} \big ( R_{1,12}(0;n)R^{(1)}_{23}(n)+R_{1,14}(0;n)R^{(1)}_{43}(n) \big )+\mathcal {O}\big ({\rm e}^{-3cn}\big ).\end{gather}Note that \mathcal {B}_{12}(n), \mathcal {B}_{32}(n) are of order {\cal O}{\big ({\rm e}^{-cn}\big )}, while \mathcal {B}_{42}(n) is of order {\cal O}{\big ({\rm e}^{-2cn}\big )}.}Revisiting (\ref {C^{\prime }s-to-P}) we have\begin{gather*}\begin{pmatrix}1 & C_1(n) & 0 & C_3(n) \\0 & C_2(n) & 1 & C_4(n)\end{pmatrix} P(n) = \begin{pmatrix}C_1(n) & 1 & C_3(n) & 0 \\C_2(n) & 0 & C_4(n) & 1\end{pmatrix}\!.\end{gather*}In particular, in view of (\ref {condition 2222}), we can write the following two equations for the constants C_2 and C_4\begin{gather*}C_2(n) P_{21}(n) + P_{31}(n) + C_4(n)P_{41}(n) = C_2(n),\\C_2(n)P_{22}(n) + P_{32}(n) + C_4(n)P_{42}(n) = 0.\end{gather*}Solving for C_2 and C_4 we find\begin{gather*}C_2(n) = \frac{P_{42}(n)P_{31}(n)-P_{41}(n)P_{32}(n)}{(1-P_{21}(n))P_{42}(n)+P_{41}(n)P_{22}(n)}, \\ C_4(n) =- \frac{P_{22}(n)P_{31}(n)+[1-P_{21}(n)]P_{32}(n)}{(1-P_{21}(n))P_{42}(n)+P_{41}(n)P_{22}(n)}.\end{gather*}From (\ref {T+H P(n) Expansion}) and (\ref {cond111}) we have\begin{gather}C_2(n)= \frac{C_{\rho }(0)}{\mathcal {E}(n) } \big ( 1+ {\cal O}{\big ({\rm e}^{-c_1n}\big )}\big ),\end{gather}and\begin{gather}C_4(n) = - \frac{2}{\alpha (0)\mathcal {E}(n) } \big ( 1+ {\cal O}{\big ({\rm e}^{-c_1n}\big )}\big ).\end{gather}Combining (\ref {T+H h_n C^{\prime }s B^{\prime }s}), (\ref {T+H Selected Elements of B}), (\ref {T+H C2 ghadim}) and (\ref {T+H C4 ghadim}) we obtain\begin{gather}h_{n-1} = -\alpha (0) \frac{\mathcal {E}(n)}{\frac{2}{\alpha (0)}R^{(1)}_{43}(n)-C_{\rho }(0)R^{(1)}_{23}(n)}\big (1+ {\cal O}{\big ({\rm e}^{-c_1n}\big )}\big ), \qquad n \rightarrow \infty .\end{gather}Note that from (\ref {T+H entries of R_1}) and (\ref {T+H Entries of the subleading term of R in expansion near zero}) we have\begin{gather*}R_{1,jk}(0;n) = R^{(1)}_{jk}(n+1), \qquad \mbox{for} \quad jk=12,14,23,43,\\R_{1,jk}(0;n) = R^{(1)}_{jk}(n-1), \qquad \mbox{for} \quad jk=21,32,34,41.\end{gather*}This allows us to rewrite (\ref {T+H h_n asymp 1}) as\begin{gather}h_{n-1}= -\alpha (0)\frac{\mathcal {E}(n)}{\mathcal {E}(n-1)}\big (1+ {\cal O}{\big ({\rm e}^{-c_1n}\big )}\big ), \qquad n \rightarrow \infty .\end{gather}This concludes the proof of Theorem~\ref {T+H main thm}.\end{gather*}Let \mathfrak {C} denote the class of symbol pairs (\phi ,d \phi ), where \phi  and d satisfy the properties mentioned in Theorem~\ref {T+H main thm}, for~which the corresponding matrix P(n) satisfies the condition~(\ref {condition 2222}) for~sufficiently large n. Now, consider the subclass \mathfrak {C}_0 \subset \mathfrak {C}, which includes symbol pairs (\phi ,d \phi ) for which~ \mathcal {E}(n) satisfies inequality (\ref {Enneq0}), or rather its generalization~(\ref {Enneq00})for sufficiently large~n (see also Section~\ref {characterization}). For the purposes of this paper, it is worthwhile to prove that the subclass~\mathfrak {C}_0 is not empty by providing an explicit example. To this end, let us consider the symbol pairs (\phi , d\phi ), where\begin{gather}\phi (z) = \left(\frac{z-b}{z-a}\right)^{\alpha },\end{gather}and\begin{gather}d(z) =\left(\frac{z-b_1}{z-a_1}\right)^{\alpha _1}\left(\frac{a_1z-1}{b_1z-1}\right)^{\alpha _1}.\end{gather}We assume that 0< a < b < a_1< b_1 < 1, and |\Re \alpha _1| < 1. Then, for \mathcal {E}(n) defined in (\ref {En000}), by~standard Watson lemma type asymptotic analysis of integrals we will arrive at\begin{gather}\mathcal {E}(n) = \kappa b_1^{n-\alpha _1}n^{\alpha _1 -1}\bigg (1 + \mathcal {O}\bigg (\frac{1}{n}\bigg )\bigg ), \qquad n \rightarrow \infty ,\end{gather}where\begin{gather}\kappa = -\frac{{\rm i}}{\pi }{\rm e}^{-{\rm i}\pi \alpha _1}\Gamma (1-\alpha _1)(b_1-a_1)^{\alpha _1}\bigg (\frac{1-bb_1}{1-ab_1}\bigg )^{\alpha }\bigg (\frac{a_1}{b_1}\bigg )^{-\alpha _1} \nonumber \\ \phantom{\kappa =}\times \bigg [{\rm e}^{-{\rm i}\pi \alpha _1}\frac{{\rm i}}{\pi }\int _{1/b_1}^{1/a_1} \bigg (\frac{z-b_1}{z-a_1}\bigg )^{\alpha _1}\bigg (\frac{1/a_1-z}{z-1/b_1}\bigg )^{-\alpha _1}\frac{z+b_1}{z-b_1}\frac{dz}{z} - 1\bigg ],\end{gather}and\begin{gather*}\arg \bigg (\frac{z-b_1}{z-a_1}\bigg ) = \arg \bigg (\frac{1/a_1-z}{z-1/b_1}\bigg ) = 0, \qquad z \in [1/b_1, 1/a_1].\end{gather*}We argue, that for generic values of a, b, a_1, b_1, \alpha  and \alpha _1 the number \kappa \ne 0. Indeed,let us take, for instance, \alpha _1 real, i.e.,\begin{gather*}-1 < \alpha _1 <1.\end{gather*}Then, the integrand in (\ref {2}) is strictly positive and hence, except for \alpha _1 = \pm 1/2, the first term in the brackets of (\ref {2}) has nonzero imaginary part and thus can not cancel the second term. It~is only needed now to notice that in the case under consideration r_0 = b_1 and hence inequality \kappa \ne 0 implies that\mathcal {E}(n) corresponding to the weights (\ref {phi example}) and (\ref {d example}) satisfies the condition~(\ref {Enneq00}) indicated in Remark \ref {Remark slight generalization} with \varkappa (n)=n^{\alpha _1-1}.}We conclude the discussion of this example by indicating the asymptotic behavior of the correspondingnorm parameter h_{n-1} and the determinant D_n. To this end we notice that, in~fact, the estimate (\ref {En asymp}) can be extended to the full asymptotic series,\begin{gather*}\mathcal {E}(n) \sim \kappa b_1^{n-\alpha _1}n^{\alpha _1 -1}\Bigg (1 + \sum _{k=1}^{\infty } d_kn^{-k}\Bigg ), \qquad n\rightarrow \infty .\end{gather*}Therefore, from (\ref {T+H h_n asymp 2}) we should have that\begin{gather*}h_{n-1} = -b_1\bigg (\frac{n}{n-1}\bigg )^{\alpha _1 -1} \bigg (1 + O\bigg (\frac{1}{n^2}\bigg )\bigg ), \qquad n \rightarrow \infty ,\end{gather*}and hence\begin{gather*}D_n = C (-b_1)^{n } n^{\alpha _1 -1}\bigg (1 + O\bigg (\frac{1}{n}\bigg )\bigg ), \qquad n \rightarrow \infty ,\end{gather*} for some constant C which is yet to be determined. It is interesting that this formulaindicates an oscillatory behavior of the determinant D_n for large n. This fact is confirmed numerically.\) 

Remark 4.1 From (), (), (REF ), and (REF ) we have the following representation for the
orthogonal polynomials \(\mathcal {P}_n(z)\)  in terms of the solution \(\mathcal {X}(z;n)\)  of the
\(\mathcal {X}\) -RHP,
\(\mathcal {P}_n(z) = \mathcal {X}_{11}(z;n) + C_1(n)\mathcal {X}_{21}(z;n) + C_3(n)\mathcal {X}_{41}(z;n).\) 
The asymptotic results concerning the function \(\mathcal {X}(z;n)\)  obtained in this section can be translated to the
large \(n\)  asymptotic formulae for the polynomials \(\mathcal {P}_n(z)\) . Indeed, skipping the rather tedious
though straightforward calculations, we arrive at the following asymptotics for \(\mathcal {P}_n(z)\) 
on the unit circle:
\(\mathcal {P}_n(z) = \frac{\beta _+(z) (2\alpha (0)C_{\rho ,+}(z) - C_{\rho }(0)\alpha (0))}{\mathcal {E}(n)} \nonumber \\\phantom{\mathcal {P}_n(z)=}{}+ \alpha _-(z)z^n \bigg ( 1 + \frac{C_{\rho }(0)\alpha (0)R_{1,21}(z;n)-2R_{1,41}(z;n)}{\mathcal {E}(n)} \bigg ) + \mathcal {O}\big ({\rm e}^{-c_1n}\big ),\) 
as \(n \rightarrow \infty \) . While in the interior and exterior of the unit circle we have the following asymptotic formulae for \(\mathcal {P}_n(z)\) :
\(\mathcal {P}_n(z)= \frac{\beta (z) ( 2\alpha (0)C_{\rho }(z) - C_{\rho }(0)\alpha (0) )}{\mathcal {E}(n)}+ \mathcal {O}\big ({\rm e}^{-c_1n}\big ), \qquad |z|<1,\\\mathcal {P}_n(z)= \alpha (z)z^n \bigg (1 + \frac{C_{\rho }(0)\alpha (0)R_{1,21}(z;n)-2R_{1,41}(z;n)}{\mathcal {E}(n)} + \mathcal {O}\big ({\rm e}^{-2c_1n}\big ) \bigg ) , \qquad |z|>1,\) 
as \(n\rightarrow \infty \)  (compare with (REF )).

Remark 4.2 
In a similar fashion, we could have obtained the large \(n\)  asymptotic expression for the norm \(h_n\)  of orthogonal polynomials (REF ). In that case, the methods of this section would work analogously if the function
\(d:= -\phi ^{-1}(z)\tilde{u}(z) \equiv \phi ^{-1}(z) \int _{a}^{b} \frac{t^{s-1}w(t)}{1-tz} \, \mathrm {d} t,\) 
satisfies \(d\tilde{d}\equiv 1\)  on the unit circle, or
\(\phi (z)\tilde{\phi }(z) = \int _{a}^{b} \frac{t^{s-1}w(t)}{1-tz} \mathrm {d} t \int _{a}^{b} \frac{t^{s-1}w(t)}{1-tz^{-1}} \, \mathrm {d} t, \qquad z\in \) 



Remaining open questions
We consider this work as a starting point of a long term research project. There are many challenging technical as well as conceptual open questions related to the Riemann–Hilbert formalism we are suggesting in this paper. Here we highlight some of them that we consider the most pressing.
Derivation of the relevant Christoffel–Darboux formulae and differential identitiesDerivation of the relevant Christoffel–Darboux formulae and differential identities
Our main objective in this paper has been to develop a \(4 \times 4\)  steepest descent analysis for certain Toeplitz+Hankel determinants and we have achieved that. However, to obtain the asymptotics of \(D_n(\phi ,d\phi ,1,1)\)  one has to derive suitable differential identities. We propose that the differential identity has to be with respect to the parameters \(\alpha _i\)  in the function \(d\)  given by (REF ). Thus, one has to perform \(m\)  integrations in the parameters \(\alpha _i\) , \(1 \le i \le m\) . Note that for \(\alpha _1 = \alpha _2 = \cdots = \alpha _m = 0\) , we have \(d \equiv 1\)  and hence \(\phi =w\) . Hence the starting point of the integration in \(\alpha _1\)  could be taken from the results of [1]}.Although the authors in [1]} do not particularly study the asymptotics of \(D_n(\phi ,\phi ,1,1)\) , this asymptotics is achievable by their methods. Integration of the differential identity in \(\alpha _1\)  will provide us with an asymptotic expression for \(D_n(\phi , d_1\phi , 1 , 1)\) , which also serves as the starting point of integration in \(\alpha _2\) . Thus we can find asymptotics of \(D_n(\phi ,d_1d_2\phi , 1 , 1)\)  which also serves as the starting point of integration in \(\alpha _3\) , and so on. Repeating this procedure will finally lead us to the asymptotics of \(D_n(\phi ,d\phi , 1 , 1)\) .
In order to derive the differential identities mentioned above, one has to find recurrence relations and prove a Christoffel–Darboux formula for the polynomials (REF ) and follow a path similar to that introduced by I. Krasovsky in [5]}. Note that the recurrence relations can be found by analyzing the function \(\mathcal {M}(z;n):= \mathcal {X}(z;n+1)\mathcal {X}^{-1}(z;n)\) , which is holomorphic in \(\lbrace 0\rbrace \)  and can be globally determined by its singular parts at zero and infinity.

Extension of the analysis to general offset values \({r,s \ne 1}\)Extension of the analysis to general offset values r,s no= 1
If we lift the restriction \(r=s=1\)  then in the jump matrix of the \(\mathcal {X}\) -RHP
the functions \(\phi (z)\)  and \(w(z)\)  should be replaced by \(z^{1-r}\phi (z)\) 
and \(z^{1-s}w(z)\) , respectively. This, in turn would raise a serious question about solvabilty
of the \(\Lambda \) -RHP. Indeed, for instance, we would not be able to define the function \(\alpha \)  in (REF ) and
hence to factorize the jump matrix (REF ). The way out of this difficulty could be the
use of the relation between the determinants \(D_n(\phi ,w;r,s)\)  with different values of \(r\) , \(s\)  or both. Such relations
are well known in the pure Toeplitz case (for example see [10]}). However, for general Toeplitz+Hankel determinants they are yet to be found.
Another way to approach the problem could be to develop the so-called Bäklund–Schlesinger transformations
of the \(\mathcal {X}\) -RHP itself. That is, the transformations of the form,
\(\mathcal {X}(z) \mapsto R(z)\mathcal {X}(z),\) 
where \(R(z)\)  is a properly chosen rational function for which the above transformation results in the desired shifting of the parameters \(r\)  and \(s\) . Also, one can try
to allow the matrix \(P(n)\)  in the setting of the \(\mathcal {X}\) -RHP to depend on \(z\) .
It is worth to point out that the problem with the extension of our scheme to the general values of \(r\)  and \(s\)  is not
actually a problem of the setting of the relevant Riemann–Hilbert problem. Indeed, it is rather the question of
the correct way to approach to its asymptotic solution. Let us demonstrate this point by considering the pure Toeplitz case.
Assume that \(D_n(\phi ,0;r,0) \ne 0\)  for all \(n\) , so that \(\mathcal {P}_n(z)\)  and, correspondingly, the solution \(\mathcal {Y}(z;n)\)  of the \(\mathcal {Y}\) -RHP exist for all \(n\) . Put,
\(\mathcal {X}(z;n):= P^{-1}_{\infty }(n)\begin{pmatrix}\mathcal {Y}_{11}(z;n)&0&0&\widetilde{\mathcal {Y}}_{12}(z;n)\\0&\widetilde{\mathcal {Y}}_{11}(z;n)&\mathcal {Y}_{12}(z;n)&0\\0&\widetilde{\mathcal {Y}}_{21}(z;n)&\mathcal {Y}_{22}(z;n)&0\\\mathcal {Y}_{21}(z;n)&0&0&\widetilde{\mathcal {Y}}_{22}(z;n)\end{pmatrix}\!,\) 
where the normalizing matrix \(P_{\infty }(n)\)  is given by
\(P_{\infty }(n)= \begin{pmatrix}1&0&0&\mathcal {Y}_{12}(0;n)\\0&\mathcal {Y}_{11}(0;n)&0&0\\0&\mathcal {Y}_{21}(0;n)&1&0\\0&0&0&\mathcal {Y}_{22}(0;n)\end{pmatrix}\!,\) 
which is invertible for generic \(\phi \) . It is straightforward to check that the so defined \(4\times 4\)  matrix-valued function \(\mathcal {X}\)  solves the \(\mathcal {X}\) -Riemann–Hilbert problem, RH-\(\mathcal {X}\) 1–RH-\(\mathcal {X}\) 4 with \(w \equiv 0\) , and \(\phi (z)\)  replaced by \(z^{1-r}\phi (z)\) . Take now \(r=0\) . That is, let us consider the standard orthogonal polynomials
on the circle with the weight \(\phi (z)\)  having zero winding number. Then, from the standard \(2\times 2\) 
Riemann–Hilbert formalism [11]}, we know everything about the asymptotic behavior of the corresponding orthogonal polynomials \(\mathcal {P}_n(z)\)  and hence we know asymptotic solution of the \(\mathcal {X}\) -Riemann–Hilbert problem corresponding
to \(D_n(\phi ,0;0,0)\) . And, we know this in spite of the fact that the approach developed in the body of this work can not be satisfactorily used for the case \(r=s=0\) . We believe that this observation might entail a useful hint on how to modify our Riemann–Hilbert approach to deal with general values of \(r\)  and \(s\) .

Extension of the Riemann–Hilbert analysis of Section  for more general choices of \({I}\)Extension of the Riemann–Hilbert analysis of Section  for more general choices of \(I\)
We recall that our Riemann–Hilbert analysis of Section , with minimal modifications, naturally extends to the following three cases as well: i) \(-1<a<b<0\) , ii) \(-\infty <a<b<-1\) , and iii) \(1<a<b<\infty \) . A natural first step in generalizing the results of Section  beyond the above cases is considering the case where \(I\)  has 0 as an end point. This would slightly affect the analysis as one has to take into account the behavior of \(w\)  at 0 in the set up of the \(2\times 4\)  and the subsequent Riemann–Hilbert problems.
The other interesting case to be studied is when \(I\)  intersects the unit circle. Clearly, in this case one has to perform local analysis in a neighborhood of the intersection point(s) of \(I\)  and the unit circle. Although these local constructions are reminiscent of what one does near the Fisher–Hartwig singularities or the endpoints of the support of the symbol, here even if the possible intersection points \(\pm 1\) , are regular points for non-FH symbols \(\phi \)  and \(w\) , one has to still perform local analysis due to collision of the supports of \(\phi \)  and \(w\) .
Another possible generalization would be to consider \(I\)  to be the union of two symmetric intervals with respect to the unit circle, i.e., \(I=[a,b] \cup \big [b^{-1},a^{-1}\big ]\) . This generalization should be accessible by slight modification of our approach explained in Section . However, generalization to the case where \(I\)  is a union of two non-symmetrical intervals with respect to the unit circle needs a more special treatment.

Extension to Fisher–Hartwig symbols
One can study the large-\(n\)  asymptotics of determinant \(D_n(\phi ,d \phi ,1,1)\)  (and with increasing effort \(D_n(\phi ,d \phi ,r,s)\)  for fixed \(r, s \in {\mathbb {Z}}\) ) assuming that \(\phi \)  possesses Fisher–Hartwig singularities \(\lbrace z_i\rbrace ^{m}_{i=1}\)  on the unit circle. It is in fact in this level of generality that E. Basor and T. Ehrhardt have been able to compute the asymptotics of \(D_n(\phi ,d \phi ,0,1)\) , \(D_n(-\phi ,d \phi ,0,1)\) , \(D_n(\phi ,d z \phi ,0,1)\) , and \(D_n(-z\phi ,d \phi ,0,1)\)  via the operator-theoretic methods in [1]}. However, the authors in [1]} further require that the Fisher–Hartwig part of \(\phi \)  be even. In fact they used some results of the work [10]} to prove their asymptotic formulas for Toeplitz+Hankel determinants, and for this reason they inherited the evenness assumption from the work [10]} where the authors needed evenness of \(\phi \)  in their \(2\times 2\)  setting to relate Hankel and Toeplitz+Hankel determinants to a Toeplitz determinant with symbol \(\phi \) .
From a Riemann–Hilbert perspective, in the presence of Fisher–Hartwig singularities, one has to construct the \(4\times 4\)  local parametrices near the points \(z_i\) . Expectedly, these local parametrices must be expressed in terms of confluent hypergeometric functions as suggested by [10]}. We have not yet worked out the details of this construction but we believe that it should be well within reach. It would be methodologically important to achieve the results obtained from operator-theoretic tools via the Riemann–Hilbert approach as well. Moreover, we expect that the evenness of the Fisher–Hartwig part of \(\phi \)  would not play a role in our \(4 \times 4\)  setting, and in that sense there are reasonable prospects of generalizing the results of [1]} to symbols \(\phi \)  with non-even Fisher–Hartwig part.

Characterization of generic classes of Szegő-type symbol pairs
\({(\phi ,d \phi )}\) , with \({d\tilde{d}\equiv 1}\)  on the unit circleCharacterization of generic classes of Szego-type symbol pairs (phi,dphi), with d tilde d equiv 1 on the unit circle
Take one of the six conditions of Lemma REF . Denote by \(\mathfrak {P}(n)\)  and \(\mathcal {E}(n)\) , respectively, the corresponding nonzero quantity and its leading order term in the large \(n\)  asymptotic expansion. Consider the class of symbol pairs \((\phi ,d\phi )\) , where \(\phi \)  and \(d\)  satisfy the properties mentioned in Theorem . Within this class, take the subclass \(\mathfrak {C}\)  of symbol pairs for which the elements of the corresponding matrix \(P(n)\)  satisfy \(\mathfrak {P}(n) \ne 0\)  for sufficiently large \(n\) . Also consider the subclass \(\mathfrak {C}_0 \subseteq \mathfrak {C}\)  of symbols pairs for which \(\mathcal {E}(n)\) 
satisfies () for sufficiently large \(n\) . One should be able to find the asymptotics of the norm \(h_n\)  of orthogonal polynomials in a similar fashion as presented in Section  in terms of \(\mathcal {E}(n)\) , assuming that \((\phi ,d\phi ) \in \mathfrak {C}_0\) . It would be very interesting to completely or partially characterize the classes of symbol pairs \(\mathfrak {C}\)  and \(\mathfrak {C}_0\) , corresponding to each one of the six conditions of Lemma REF .
Moreover, although we have provided explicit examples for a class of Szegő-type functions \(d\)  which further satisfy \(d\tilde{d}\equiv 1\)  on the unit circle (see the beginning of Section ), a complete characterization of such functions is currently unknown to the authors. Also, regarding what we discussed in Section , for a given Szegő-type symbol \(\phi \) , we are very interested to find the associated class of weights \(w\) , supported on the interval, for which the equality (REF ) holds.

Characteristic polynomial of a Hankel matrix
As mentioned in the Introduction, arguably the most important motivation behind studying the asymptotics of Toeplitz+Hankel determinants is to study the large \(n\)  asymptotics of the eigenvalues of the matrix \(H_n[w]\) . We recall that the characteristic polynomial \(\det (H_n[w]-\lambda I)\)  is indeed the Toeplitz+Hankel determinant \(D_n(-\lambda ,w,0,0)\) . In this case the associated \(\Lambda \) -model Riemann–Hilbert problem needs a special treatment. In a sense it is a simpler problem as the symbol \(\phi \)  is identically equal to a constant, but more complicated – compared to the situation in Section  – as it does not enjoy \(J_{\Lambda ,23}(z)=0\) . In any case, the solution to this model problem provides us with the constant term in the asymptotics of \(D_n(-\lambda ,w,0,0)\) , and in the case of Fisher–Hartwig weight \(w\) , one can hope to obtain the leading terms of this asymptotic expansion (up to the constant term, viz. the solution of the \(\Lambda \) -model problem) from the local analysis near the Fisher–Hartwig singularities.The very recent work [18]} shows that indeed one can explicitly describe the leading terms of the asymptotics of \(D_n(-\lambda , w, 0,0)\) , in the case of \(w\)  having jump discontinuites and \(\lambda \)  sufficiently large. This last point is yet another motivation to pursue the goals of Section REF .

Acknowledgements
We are very grateful to Estelle Basor, Thomas Bothner, Christophe Charlier, Percy Deift and Igor Krasovsky for their interest in this work and for many very useful comments and suggestions. We also thank the referees for their valuable remarks. R. Gharakhloo acknowledges support by NSF-grant DMS-1700261. A. Its acknowledges support by NSF-grant
DMS-1700261 and by Russian Science Foundation grant No. 17-11-01126.
[1]Referencesref
